# How do you find the domain of inverse trig functions?

The domain of any inverse trig function ($\arcsin$, $\arccos$, $\arctan$) is equal to the range of the corresponding trig function ($\sin$, $\cos$, $\tan$).
So the domain of $\arcsin$ and $\arccos$ is $\left\{x \in \mathbb{R} : - 1 \le x \le 1\right\}$ since $- 1 \le \sin \theta \le 1$ and $- 1 \le \cos \theta \le 1$ for all $\theta \in \mathbb{R}$.
The domain of $\arctan$ is $\mathbb{R}$ since the range of $\tan \theta$ is the whole of $\mathbb{R}$.
The domain of ${\sec}^{-} 1$ and ${\csc}^{-} 1$ is $\left\{x \in \mathbb{R} : x \le - 1 \mathmr{and} x \ge 1\right\}$
The domain of ${\cot}^{-} 1$ is $\mathbb{R}$.